Homework chapter 6:
Do 5 of the following exercises (due by 4/10)
1. Prove Theorem 0.1
Theorem 0.1. Let a, b and c be integers. (a) If a ̸= 0, a|a.
(b) If a|b and b|a then b=±a.
(c) If a|b and b|c then a|c.
2. Prove that the following are true for every integer n: (a) 2 divides n(n + 1).
(b) 3 divides n(n + 1)(n + 2).
(c) 4 divides n(n + 1)(n + 2)(n + 3).
(d) 6 divides n3 − n.
(e) 9 divides n3 +(n+1)3 +(n+2)3.
3. Provethatforalln∈N,64divides9n −8n−1.
4. Prove that for every n ∈ Z, 3|n if and only if 3|n2.
5. Prove that 3 does not have a square root in Q.
6. Prove the uniqueness in Theorem 0.2.
Theorem 0.2. (the Euclidean algorithm)
Let a, d ∈ Z with d > 0. There exist unique integers q and r such that
(i) a = qd + r (ii) 0≤r
(b) Let d be the least element of S (why does d exist?). Prove that d|a and d|b. (c) Let c ∈ N be such that c|a and c|b. Prove that c|d.
(d is called the greatest common divisor of a and b. Notice that d is an element of S, so there exist m, n ∈ Z such that d = ma + nb.)
8.Letpbeaprimeinteger,leta∈Zwitha>0,andsupposethatp∤a. Provethatthe greatest common divisor of p and a equals 1.
9. Let p be a prime integer, let a, b ∈ Z, and suppose that p|ab. Prove that p|a or p|b.
10. Let p and q be prime integers with p ̸= q. Let n ∈ Z and suppose that p|n and q|n. Prove that pq|n.
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